Second Moment Convergence Rates for Uniform Empirical Processes
© Y.-Y. Chen and L.-X. Zhang. 2010
Received: 21 May 2010
Accepted: 19 August 2010
Published: 23 August 2010
1. Introduction and Main Results
if and . The converse part was proved by the study of Erdös in . Obviously, the sum in (1.1) tends to infinity as . Many authors studied the exact rates in terms of (cf. [3–5]). Chow  studied the complete convergence of , . Recently, Liu and Lin  introduced a new kind of complete moment convergence which is interesting, and got the precise rate of it as follows.
Other than partial sums, many authors investigated precise rates in some different cases, such as U-statistics (cf. [8, 9]) and self-normalized sums (cf. [10, 11]). Zhang and Yang  extended the precise asymptotic results to the uniform empirical process. We suppose is the sample of random variables and is the empirical distribution function of it. Denote the uniform empirical process by , , and the norm of a function on by . Let , be the Brownian bridge. We present one result of Zhang and Yang  as follows.
Inspired by the above conclusions, we consider second moment convergence rates for the uniform empirical process in the law of iterated logarithm and the law of the logarithm. Throughout this paper, let denote a positive constant whose values can be different from one place to another. will denote the largest integer . The following two theorems are our main results.
Consequently, explicit results of (1.4) and (1.5) can be calculated further.
2. The Proofs
In order to prove Theorem 1.1, we present several propositions first.
The calculation here is analogous to (2.1), so it is omitted here.
Proof of Theorem 1.1.
Proof of Theorem 1.2.
Combine (2.22), (2.23), and (2.24)together, we get the result of Theorem 1.2.
This work was supported by NSFC (No. 10771192) and ZJNSF (No. J20091364).
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